We have discussed some theoretical results for spin ice on arbitrary graphs. In the first part of the paper we have provided a graph theoretical mapping between spin ices and Ising models based on the directed incidence matrix of the Line graph. Specifically, we have shown something that was already known: while all spin ices can be mapped to Ising models, not all Ising models can be mapped to spin ices; but here we show a general a procedure to extract the original spin ice. We have also shown that all spin ices are degenerate (except the trivial case: the 1D Ising model). We have proved this result using the known gauge transformation for the Ising model. Another fact which we managed to prove in the first part of this paper, and that we found surprising, is that for planar spin ices, degeneracy is only at the vertex level. This implies that the effective Ising model, cycles between different vertices of the original spin ice are unfrustrated, as it can be seen via a gauge transformation to align these vertices. The same is not true for the interactions at the vertices, implying that the frustration scales with the number of vertices of the original spin ice. In the second part of the paper we have focused on techniques to bound the degeneracy of the ice manifold. The method is based on the number of Eulerian tours, as for every balanced graph (ice manifold state) there exists at least one Eulerian trail. The advantage of using the Eulerian trails is that there are spectral techniques to count the number of Eulerian trails of a directed graph. We applied these techniques for even-degree and connected graphs, showing that these bounds can be extended to the case of odd-degree reducible graphs. To conclude, we have used an exact formulation for the degeneracy of spin ices configurations using a permanent identity. Given the fact that the permanent is \(\#P\) -complete quantity to compute, we have employed numerical methods based on the Bethe free energy. Such method provides numerical lower bounds to the permanent, and we have thus obtained lower bounds to the spin ice degeneracy for various spin ice regular lattices. The advantage of such procedure is that these certified lower bound estimates can be obtained relatively quickly and without using loop Monte Carlo techniques [1]}, [2]}, [3]} which, however, would give a more precise estimate.
